The question of pattern selection in the presence of noise is addressed in the context of the one-dimensional Swift-Hohenberg equation, a model for the onset of convection. We show how noise destroys long-range order in the long-time patterns, so that characterization of the selected pattern in terms of the Fourier mode with the maximum spectral power is not always suitable. The number of zeros of the configurations turns out to be a better quantity. We consider also the decay process after an Eckhaus instability. It is shown that the selected pattern is close to the one of fastest growth during the linear regime, and not to the variationally preferred. This mechanism is robust to small noise.
Bibliographical noteFunding Information:
This work has been supported by NATO, within the program "Chaos, order and patterns; aspects on nonlinearity", project number 890482, by the Supercomputer Computations Research Institute, which is partially funded by the US Department of Energy contract No. DE-FC05-85ER25000, by the Direcci6n General de Investigaci6n Cientifica y T6cnica, contract number PB 89-0424, and by the Universitat de les Illes Balears. Most of the calculations reported here have been performed in the 64k-node Connection Machine at the Supercomputer Computations Research Institute. We are indebted to Dr. Ken Elder and Dr. Martin Grant for many useful discussions.